Given:

• $DE = x - 4$
• $EF = 2x + 5$
• $DF = 4x - 8$

Since point $E$ is between $D$ and $F$, we have: $DF = DE + EF$ Substitute the given values: $4x - 8 = (x - 4) + (2x + 5)$ Simplify the right side: $4x - 8 = x - 4 + 2x + 5$ Combine like terms: $4x - 8 = 3x + 1$ Subtract $3x$ from both sides to isolate $x$: $4x - 3x - 8 = 1$ $x - 8 = 1$ Add 8 to both sides: $x = 9$

Now that we have $x = 9$, we can find the lengths of each segment:

1. $DE$: $DE = x - 4 = 9 - 4 = 5$

2. $EF$: $EF = 2x + 5 = 2(9) + 5 = 18 + 5 = 23$

3. $DF$: $DF = 4x - 8 = 4(9) - 8 = 36 - 8 = 28$

Answer:

The lengths of the segments are:

• $DE = 5$
• $EF = 23$
• $DF = 28$

Would you like further clarification on this solution, or have any other questions?

Here are 5 related questions you might be interested in:

1. How do you set up an equation when a point lies between two other points?
2. What happens if the value of $x$ in such problems turns out to be negative?
3. How do you find the midpoint of a segment given the endpoints?
4. How do you apply the segment addition postulate in different geometric problems?
5. What are the properties of line segments in geometric figures?

Tip: Always double-check each step when solving equations involving segments to ensure accuracy.